How to Determine the Number of Electrons With Quantum Numbers

By Kylene Arnold; Updated May 23, 2018

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Describing the states of electrons in atoms can be a complicated business. Like if the English language had no words to describe orientations like "horizontal" or "vertical," or "round" or "square," a lack of terminology would lead to many misunderstandings. Physicists also need terms to describe the size, shape and orientation of the electron orbitals in an atom. But instead of using words, they use numerals called quantum numbers. Each of these numbers corresponds to a different attribute of the orbital, which allows physicists to identify the exact orbital they want to discuss. They are also related to the total number of electrons an atom can hold if this orbital is its outer, or valence, shell.

TL;DR (Too Long; Didn't Read)

TL;DR (Too Long; Didn't Read)

Determine the number of electrons using quantum numbers by first counting the number of electrons in each full orbital (based on the last fully-occupied value of the principle quantum number), then adding the electrons for the full subshells of the given value of the principle quantum number, and then adding two electrons for each possible magnetic quantum number for the last subshell.

Count the Full Orbitals

Subtract 1 from the first, or principle, quantum number. Since the orbitals must fill in order, this tells you number of orbitals that must already be full. For example, an atom with the quantum numbers 4,1,0 has a principal quantum number of 4. This means that 3 orbitals are already full.

Add the Electrons for Each Full Orbital

Add the maximum number of electrons that each full orbital can hold. Record this number for later use. For example, the first orbital can hold two electrons; the second, eight; and the third, 18. Therefore the three orbitals combined can hold 28 electrons.

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Identify the Subshell Indicated by the Angular Quantum Number

Identify the subshell represented by the second, or angular, quantum number. The numbers 0 through 3 represent the "s", "p," "d" and "f" subshells, respectively. For example, 1 identifies a "p" subshell.

Add the Electrons from the Full Subshells

Add the maximum number of electrons that each previous subshell can hold. For example, if the quantum number indicates a "p" subshell (as in the example), add the electrons in the "s" subshell (2). However, if your angular quantum number was "d," you'd need to add the electrons contained in both the "s" and "p" subshells.

Add the Electrons from Full Subshells to Those From Full Orbitals

Add this number to the electrons contained in the lower orbitals. For example, 28 + 2 = 30.

Find the Legitimate Vales for the Magnetic Quantum Number

Determine how many orientations of the final subshell are possible by determining the range of legitimate values for the third, or magnetic, quantum number. If the angular quantum number equals "l," the magnetic quantum number can be any number between "l" and " −l," inclusive. For example, when the angular quantum number is 1, the magnetic quantum number may be 1, 0 or −1.

Count the Number of Possible Subshell Orientations

Count the number of possible subshell orientations up to and including the one that is indicated by the magnetic quantum number. Begin with the lowest number. For example, 0 represents the second possible orientation for the sublevel.

Add Two Electrons Per Possible Orientation to the Previous Sum

Add two electrons for each of the orientations to the previous electron sum. This is the total number of electrons an atom can contain up through this orbital. For example, since 30 + 2 + 2 = 34, an atom with a valence shell described by the numbers 4,1,0 holds a maximum of 34 electrons.

About the Author

Kylene Arnold is a freelance writer who has written for a variety of print and online publications. She has acted as a copywriter and screenplay consultant for Advent Film Group and as a promotional writer for Cinnamom Bakery. She holds a Bachelor of Science in cinema and video production from Bob Jones University.